Divide the following complex numbers. $ \dfrac{-5-37i}{-3+5i}$
Solution: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-3-5i}$ $ \dfrac{-5-37i}{-3+5i} = \dfrac{-5-37i}{-3+5i} \cdot \dfrac{{-3-5i}}{{-3-5i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(-5-37i) \cdot (-3-5i)} {(-3+5i) \cdot (-3-5i)} = \dfrac{(-5-37i) \cdot (-3-5i)} {(-3)^2 - (5i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(-5-37i) \cdot (-3-5i)} {(-3)^2 - (5i)^2} = $ $ \dfrac{(-5-37i) \cdot (-3-5i)} {9 + 25} = $ $ \dfrac{(-5-37i) \cdot (-3-5i)} {34} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({-5-37i}) \cdot ({-3-5i})} {34} = $ $ \dfrac{{-5} \cdot {(-3)} + {-37} \cdot {(-3) i} + {-5} \cdot {-5 i} + {-37} \cdot {-5 i^2}} {34} $ Evaluate each product of two numbers. $ \dfrac{15 + 111i + 25i + 185 i^2} {34} $ Finally, simplify the fraction. $ \dfrac{15 + 111i + 25i - 185} {34} = \dfrac{-170 + 136i} {34} = -5+4i $